The answer you want depends on how deep you want to go I suppose. Of primary interest is the intrinsic magnetic moment. This makes a contribution to the magnetic field but does not result from the motion of charge around any point. That means the source of this magnetism is different from an electromagnet, which only works when electricity is flowing.

The intrinsic magnetism of the electron is connected to its intrinsic angular momentum, known as spin.

Spin relates to the Pauli exclusion principle. To really understand it you have to learn quantum mechanics.

Also, there is another portion of the magnetic field which does come from the electrons motion. It's not intrinsic, but it can be thought of in terms of moments. The whole idea of moments can be tied into expanding functions in a basis of spherical harmonics. But often what is meant by magnetic moment is not this, but rather the intrinsic moment due to spin only.